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Compensating for irregular shot illumination

 In the first section of this Chapter, I discussed how shot-profile migration is the adjoint of a linear first-order Born modeling operator. Given such a modeling operator, geophysical inverse theory [e.g. Tarantola (1987)] provides a rigorous framework which allows us to estimate a model of the subsurface that is unbiased by our recording geometry and the effects of wave propagation. Several authors [see Ronen and Liner (2000) for a full review] leverage this, and calculate a least-squares (L2) pseudo-inverse to the forward modeling operator.

For a generic forward modeling operator, ${\bf A}$, and more data values than unknown model parameters, the model estimate, $\hat{\bf m}$ that best predicts the data in a least-squares sense is given by the solution to the normal equations,  
{\bf A}' \, {\bf A} \; {\hat {\bf m}} = 
{\bf A}' \, {\bf d}.\end{displaymath} (82)

Often for large geophysical problems such as migration, this system of equations is not solved directly, rather the solution is estimated by an iterative method such as conjugate gradients [e.g. Nemeth et al. (1999); Prucha et al. (2000)].

Obviously, however, if the operator, ${\bf A}' \,{\bf A}$ is diagonal, it is easy to calculate $({\bf A}' \, {\bf A})^{-1}$, and obtain the L2 solution directly from the adjoint (migrated) image. Rather than iterate with the entire migration operator, in this section I will consider the effect of explicitly inverting the chain of operators that make up wave-equation modeling, looking for situations in which ${\bf A}' \,{\bf A}$ is diagonal.

The shot-profile modeling operator, given in equation ([*]), can also be expressed in terms of the extrapolation operator, ${\bf B}$, as
{\bf d} & = & {\bf A}_{\rm SP} \; {\bf m} \\ & = &
{\bf B} \; 
 ..._{-} \; 
{\bf \Sigma}_{\omega}' \; 
{\bf \Sigma}_{s}' \; 
{\bf m}.\end{eqnarray} (83)

The classical L2 estimate of ${\bf m}$ is given by
{\hat {\bf m}} 
& = &
{\bf A}'_{\rm SP} \; {\bf A}_{\rm ...
 ...\bf \Sigma}_{s}' \; 
\right)^{-1} \;
{\bf A}'_{\rm SP} \; {\bf d}.\end{eqnarray} (85)
Following the discussion in the previous section, I will assume ${\bf B}' \; {\bf B} \approx {\bf I}$, so that
{\hat {\bf m}}
& = &
{\bf \Sigma}_{s} \; 
{\bf \Sigma}_{...
 ...d} \\ & = &
{\bf V}_{\rm SP}^{-1} \;
{\bf A}'_{\rm SP} \; {\bf d}.\end{eqnarray} (87)
Close inspection of ${\bf V}_{\rm SP} (={\bf \Sigma}_{s} \; 
{\bf \Sigma}_{\omega} \; 
\vert{\bf Q}_{-}\vert^2 \; 
{\bf \Sigma}_{\omega}' \; 
{\bf \Sigma}_{s}')$reveals that it is a diagonal matrix that can be applied as a model-space weighting function after migration. In physical terms, the diagonal of ${\bf V}_{\rm SP}$ contains the total shot illumination: the integration over frequency of $\vert{\bf q}_{-}\vert^2$, the energy in the downgoing (shot) wavefield at each model point. Shot illumination can be computed cheaply during the migration process.

This weighting function is also equivalent to the upgoing/downgoing wavefield imaging condition originally proposed by Claerbout (1971). Field data results Jacobs (1982) show this imaging condition is very susceptible to noise. However, dividing by the downgoing wavefield after migration, rather than directly as part of the imaging condition, has a significant advantage: the choice of appropriate local smoothing and stabilization parameters (e.g. $\epsilon$) may be made after the migration is finished.

Duquet et al. (2000) calculate the subsurface illumination for Kirchhoff migration by summing all contributions from a single scatterer that get modeling into dataspace. The shot illumination described here is a special case of Duquet's illumination that assumes all scattered energy is recorded. However, the extra cost of calculating shot illumination is negligible compared to the cost of a single migration, and Duquet's approach to calculating illumination is not appropriate for wave-equation migration schemes (see Appendix [*]).

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